Distributed Fault Detection for Precise and Robust Local Positioning

Reimar Pfeil, Linz Center of Mechatronics (LCM) GmbH

Klaus Pourvoyeur, Christian Doppler Laboratory for Integrated Radar Sensors, Johannes Kepler University (JKU)

Andreas Stelzer, Institute for Communications and Information Engineering (ICIE), JKU Günter Stelzhammer, ABATEC Electronic AG

BIOGRAPHY

Reimar Pfeil was born in Austria, 1980. He received the Dipl.-Ing. (M.Sc.) degree in mechatronics from the Johannes Kepler University Linz, Austria, 2007. Since 2007 Mr. Pfeil is employed at the LCM, Linz, Austria and

since 2008 he is working on his Ph.D. thesis with the topic of improving the local positioning measurement (LPM) system. His research work stretches from acoustic radar to ultra wideband signal processing with the current focus on local positioning. D.I. Pfeil has authored three conference papers and co-authored one journal paper.

Klaus Pourvoyeur was born in Austria, 1977. He received the Dipl.-Ing. (M.Sc.) degree in mechatronics and the Dr.techn. (Ph.D.) degree with highest honors from the JKU Linz, Austria, in 2005 and 2007, respectively.

He was with the ICIE at the JKU, and joined the LCM in 2004 for his Ph.D. thesis. Since 2007 Mr. Pourvoyeur is employed at the Christian Doppler Laboratory for Integrated Radar Sensors at the University of Linz, Austria. His research work focuses on position estimation, tracking, and sensor fusion within multi target scenarios for cooperative as well as non-cooperative target. Dr. Pourvoyeur has authored or co-authored nearly 20 journal as well as conference papers and works as reviewer for international journals and conferences.

Andreas Stelzer (M’00) was born in Austria, 1968. He received the Dipl.-Ing. (M.Sc.) degree in electrical engineering from the Technical University of Vienna, Austria. In 2000 he received the Dr.techn. degree (Ph.D.) in mechatronics with honors ’sub

auspiciis praesidentis rei publicae’ from the Johannes Kepler University Linz. In 2003 he became an associate professor the ICIE. Since 2007 he is head of the Christian Doppler Laboratory for Integrated Radar Sensors. His research work focuses on microwave sensor systems for industrial and automotive applications, RF- and microwave subsystems, SAW sensor systems and applications, as well as digital signal processing for sensor signal evaluation. Dr. Stelzer has authored or co-authored more than 180 journal and conference papers. In 2008 he received the IEEE-MTT Outstanding Young Engineer Award from the Microwave Theory and Techniques society.

Günter Stelzhammer was born in Austria, 1967. He received the Ing. degree 1986 in electrical engineering from the technical high school in Braunau and the Dkmf. (M.B.A.) in 2006 from the University of Passau, Germany. From 1987 to 1992 Mr.

Stelzhammer was technical engineer at the WETRON Corporation in Rosenheim, Germany and from 1992 to 1997 he worked as technical salesman for the GIFAS Electric Corporation in Eugendorf, Austria. Furthermore, he has been technical director of the DETO Automatierungstechnik from 1997 to 2002 in Kufstein, Austria and since 2006 he is business unit leader of the LPM system at Abatec Electronic in Regau, Austria.

ABSTRACT For precise and robust positioning it is of utmost importance to detect faulty measurements before they can distort the position solution. This contribution discusses the distributed fault detection of a time difference of arrival (TDOA) local positioning system by using a Kalman filter. A chi-squared test is used for the data evaluation purposes of each individual measurement separately before the measurement data are fused to calculate a position solution. The derived input estimator (IE) is based on a linear model and therefore easy to monitor for false data. The applicability of the presented algorithm to real measurement data is demonstrated. 1) INTRODUCTION Many technical applications require precise knowledge of position information. Equally important as the position solution, is robustness against various distortions caused by multi-path or shadowing effects. A popular tool for calculating a position solution is the Kalman filter (KF). A Kalman tracking filter efficiently calculates state estimations and also provides the possibility of fault detection after a settling phase. During the initialization and settling phase, the filter is extremely sensitive even to a very limited number of faulty measurements. This is severely aggravated if the measurement equations are nonlinear and consequently linearization techniques must be implemented. Therefore, immediate knowledge of the quality of all available measurements is key to calculating precise and robust state estimates. This contribution presents methods of determining the quality of measurements for the TDOA-based local position estimation system LPM to avoid corrupting the position solution. A brief discussion of the LPM system is given in the next section. The error assignment for the LPM system is presented in (Pourvoyeur, Stelzer and Stelzhammer, 2008). However, the described techniques require specific test conditions, for instance, instantaneous knowledge of the measurement transponder (MT). During normal mode of operation, this knowledge is a priori not available. Robust positioning for the LPM system is also presented in (Pfeil, 2009) but only for static position estimation and with the trade-off of increased variance. The methods presented in this publication rely on distributed monitoring of the measurements without the need for data exchange to make quality assessments. The calculated quality information is used by a subsequent fusion process of the measurements to determine the position of the MT. 2) LPM SYSTEM This section summarizes the basic system concept and discusses the LPM measurement equation.

Furthermore, the state transition modeling is briefly presented.

a) LPM CONCEPT The main concept of the LPM system is introduced in (Stelzer, Pourvoyeur and Fischer, 2004). With measurement rates of up to 1000 position solutions per second and an accuracy in the range of some centimeters, it is at the moment one of the most sophisticated local positioning systems available on the market. The LPM system operates in the license-free 5.8 GHz ISM band and is based on the well-known frequency-modulated continuous-wave (FMCW) radar principle (Stove, 1992). Applications of the LPM include tracking cars on a race track, stunt pilots during acrobatic flights, cows for animal behavior research, soccer players for training optimization and many more. The basic arrangement of an LPM system consists of base stations (BSs) surrounding the field of view. First of all, a specific BS, denoted as master BS, uses a control telegram to activate a reference transponder (RT) at a fixed and precisely known position. The RT periodically emits its signal to synchronize all BSs and therefore provides a common time base. After all BSs are synchronized, the master BS triggers a designated MT. In a multi-MT scenario, the activation of the MTs switches from one to the next with each measurement cycle. This time-sharing guarantees that multiple MTs do not disturb each other. Data from all BSs are collected by data transfer network and passed to a master processing unit (MPU), where the position solution is calculated. A sketch of the basic LPM system arrangement is shown in Fig. 1.

Figure 1 - Basic LPM system arrangement consisting of several BSs surrounding the field of view, a master BS for activating a specific MT and an RT for implementing a differential measurement principle. The data are collected via data transfer network for processing by an MPU.

b) LPM MEASUREMENT EQUATION Each BS first receives the chirp signal from the RT to which it synchronises its internally generated chirp. If the BS is the master BS, it then triggers the MT. Thus, the start time of the measurement chirp tMT is always larger than tRT. According to the FMCW measurement principle, each BS measures the frequency differences ΔfBS[n] between the signals of the RT and the MT, which can be converted into a TDOA measurement tTDOA,BS[n] by

][BSr

r][BSTDOA, nn f

BTt Δ= , (1)

with Tr = 500 μs as the ramp duration, and Br = 150 MHz as the used bandwidth. By multiplying the TDOA measurement (1) with the velocity of propagation of the electromagnetic wave c0, one obtains ρBS[n], referred to as the pseudo-range, ].[BSTDOA,0][BS c nn t=ρ (2) The complete signal map of the master BS[n], where n is the nth element in {1, …, N} and N is the total number of BSs, is depicted in Fig. 2. The measurement equation taking into account the geometry of the arrangement is given by

[ ]

ρρ

ρρ

wnnn

N

+−−−=

=

RT][BSMT][BS]BS[

T]BS[]1BS[BS

zzzz

ρ L,

(3) with zBS[n] as the Cartesian vector position of BS[n], zRT as the position of the RT, zMT as the position of the MT, wρ as a measurement offset, ||·|| as the Euclidian norm and [·]T as the transpose of a vector. A mechanical interpretation of the LPM measurement equation is given in (Stelzer, Pourvoyeur and Fischer, 2004).

Figure 2 - Signal map of the master BS of the LPM system. After the BS synchronizes its internally generated chirp to the RT, it triggers the MT and evaluates the time tTDOA,BS[n] via the FMCW principle.

c) STATE MODELS IN LPM Basically, two models have to be considered: a model for the motion of the MT and a second model for the measurement offset wρ. Advantageously, these two models are decoupled from each another. A detailed discussion of basic motion models suitable for describing the movement of an MT is given in (Li and Jilkov, 2003). Due to the high sample rate of the LPM system, decoupled models for each Cartesian coordinate axis are sufficient. Potential model candidates are a random walk (RW), a nearly constant

velocity (CV), or a nearly constant acceleration (CA) model. Using a jerk-based model would, result in an over-modeling of the state vector and consequently in a reduced performance of the state estimation. The local clocks used for timing purposes of the RT and the MT are derived from different oscillators. Hence, the point in time where the MT starts to transmit its chirp signal is influenced by the frequency difference between both oscillators. The nearly constant frequency difference results in a nearly constant drift of wρ. Therefore, the state transition of wρ is modeled adequately by a nearly CV model resulting in

kk

k

kuw

wTww

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

+ ρρ

ρ

ρ

ρ 010

1

1&&

, (4)

with Tk as the duration between MT activation at time index k and time index k+1. Possible variation in the assumption of a constant drift is accounted for by an unknown noise term uρ with zero mean and Gaussian probability density function (PDF). Although uρ is unknown, it is assumed that its statistical properties are known. 3) KALMAN-BASED STATE ESTIMATION A KF is a powerful tool to estimate the development of a state vector over time for linear and Gaussian problems. The next subsection summarizes the basic principle of a Kalman tracking filter and discusses both its usability for position estimation and tracking the measurements of a single BS. Furthermore, the necessity of a proper initialization is pointed out.

a) KALMAN TRACKING FILTER For implementing a Kalman tracking filter, a state transition equation as well as a measurement equation with respect to the state vector must be known. In case one or both equations are nonlinear, linearization techniques have to be applied. A detailed discussion of the Kalman equations with focus on their practical implementation is given in (Grewal and Andrews, 2001). The usability of a KF for tracking applications is shown in (Brookner, 1998) and (Blackman and Popoli, 1999). A KF is not designed to accept faulty measurements. Due to its infinite impulse response (IIR) characteristic, errors will persist for some time and it is therefore of utmost importance to detect and refuse anomalous sensor data.

b) POSITION ESTIMATION The state vector θMT of the Kalman tracking filter for position estimation is given by

[ ] TTMT

TMTMT )()( ρρ ww &K&zzθ = ,

(5)

and consists of the MT position and possibly higher derivatives as well as the necessary states for modeling the measurement offset. Due to the nonlinear structure of the LPM measurement equation (3), linearization techniques have to be applied (Pourvoyeur, 2007). An extended Kalman filter (EKF) approximates the measurement matrix Mk by the Jacobian of the measurement equation. For the position estimation problem of LPM Mk of BS[n] is approximated by

.1

])([

MT]1[BS

]1[BSMT

MT]1[BS

]1[BSMT

MT]1[BS

]1[BSMT ]3[]3[]2[]2[]1[]1[

TTMT

n][

⎥⎦⎤

⎢⎣⎡=

∂

∂≈

−

−

−

−

−

−

zzzzzz

zM

zzzzzz

BSk

wρ

ρ

(6) Alternatively the nonlinearities of the measurement equation can be handled by an unscented Kalman filter (UKF) as presented in (Simon, 2006). In any case the linearization depends directly on the current state estimate of the position of the MT.

c) TRACKING A SINGLE BS To set up fault detection for each BS separately, it is necessary to track the measurement output before merging the data from the BSs. The output of a BS, denoted by wBS[n], is given by

.RT]BS[MT][BS

][BS][BS

ρ

ρ

w

w

nn

nn

+−−−=

=

zzzz (7) The impact of the MT position and the measurement offset are no longer considered separately, as both are needed to calculate wBS[n] for the position estimation. The derivative of wBS[n] with respect to time is given by

(( )

)

.MTMT][BS

T][BSMT

RT][BSMT][BS

][BS][BS

ρ

ρ

ρ

w

wt

tw

n

n

nn

nn

&&

&

+−

−=

+−−−∂∂

=

∂∂

=

zzz

zz

zzzz

(8) Hence, the total velocity of the output of a specific BS depends on the radial velocity of the MT with respect to its relative position to the BS and the rate of change of wρ. For any practical application, the object velocity is much smaller than to the offset drift . (9) ρw&& <<|||| MTz

Furthermore, the geometry change from one measurement to the next is very small, due to the high measurement rates of the LPM system, and almost independent of the current model assumption of the MT motion model, resulting in

( ) ( ).

1MT][BS

T][BSMT

MT][BS

T][BSMT

+⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

−≈

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

−

kn

n

kn

n

zz

zz

zz

zz

(10) Thus, a proper model assumption for the measurement output of each BS is given by a CV model

.0

101

][BS][BS

][BS

1][BS

][BS

knkn

nk

kn

n

uwwT

ww

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

+&&

(11) Any model mismatch is taken into account by the model driving noise vector uBS[n] for BS[n]. In contrast to the measurement offset model of (4), the rate of change is no longer identical for all BSs due to the possible impact of the velocity of the MT. Both the measurement equation of (7) and the state transition equation of (11) are linear. Hence, no linearization techniques are required for tracking the output of a single BS. The state vector θBS[n] for tracking the output of each BS by a KF consist of only two states

[ ] .T

]BS[]BS[]BS[ nnn ww &=θ (12)

d) INITIALIZATION OF THE KALMAN

FILTER In Kalman-based state estimation, the initialization of the state vector and the corresponding covariance matrix is a very critical task, especially if state-dependent linearization techniques have to be applied. In position estimation, an additional solver for generating a starting solution is necessary. A discussion of various position estimation techniques for the LPM system is given in (Pourvoyeur, Stelzer and Gassenbauer, 2006). Furthermore, a robust position estimator is presented in (Pfeil, 2009). In all cases, calculating a position solution of the MT position requires the measurements from several BSs to be combined. Without additional information on the quality of the measurements, which is usually unavailable, such a solver can check the measurements of all available BSs only for consistency and must be applied before Kalman-based fault detection is in place. A serious problem arises if one or more BSs provide faulty measurements that go unnoticed. For a bad error propagation constellation of the BSs, the position estimate is too inaccurate to calculate a proper initialization value for the KF. Consequently, the filter will not be able to establish stable tracks.

Unfortunately, the initialization of a KF for tracking the output of a single BS raises the same issues, and therefore great care has to be taken during the initialization process. Anomalous values of wBS[n] may inhibit the base station Kalman filter (BS-KF) from tuning in or, even worse, may result in the tracking of wrong data and discarding of correct values. To avoid these problems, a fault detection method has to be implemented even before the BS-KF. 4) FAULT DETECTION The model for the LPM measurement equations allows fault detection by a χ2 test. Note that a χ2 test can be applied only if the error covariance matrix of the KF has reached a steady state and so the KF is in track. As mentioned before, the KF is very sensitive to outliers in this phase and it is of utmost importance to prevent faulty data from being processed. Therefore, another fault detection method has to be used to guarantee the highest possible robustness. For this purpose, a least median squares (LMS) estimator was adapted to filter false data from a given array. Only to these filtered data is the initialization of the KF applied. Note that during the initialization phase, the BS is logged off the MPU and does not forward any measurement data.

a) LMS BASICS The LMS estimator is presented in (Rousseeuw and Leroy, 2003). It is the most robust estimator and can handle up to 50% of false data. Instead of minimizing the sum of the squared residuals, i.e., the well-known least squares estimator, the LMS minimizes the median of the squared residuals such that (13) },{medianargˆ min rrθ

θo=

where is an estimate of a parameter vector θ, r is the residual vector and ‘○‘ is the Hadamard product. The median in (12) is defined for a vector x containing N elements, as

θ̂

even, is

uneven is

12

~2

~21

21~

}median{N

NNN

N

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦⎤

⎢⎣⎡ ++⎥⎦

⎤⎢⎣⎡

⎥⎦⎤

⎢⎣⎡ +

=xx

xx

(14) where x~ is the sorted variant of x. The main issue of the LMS is that there is no closed-form function describing (13) and thus, all possible values of the parameter vector θ have to be evaluated. Therefore

)!(!

!pNp

NpN

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ , (15)

evaluations have to be made, where p is the dimension of the parameter vector. It is obvious that this method becomes computationally unfeasible if N is very large. To solve this problem, Rousseeuw proposed using a set of m random subsamples determined by (16) ,))1(1(1 mpε−−− where ε is the proportion of false data and the resulting probability of having at least one good subsample is 95% (≈ 2 σ of the assumed Gaussian PDF) or more. For a state (parameter) vector given in (12) where p = 2 and a fraction ε = 0.5 of contaminated data is expected, m evaluates to 11. This makes the LMS applicable even for real time processing.

b) APPLICATION OF THE LMS ESTIMATOR TO SINGLE BS TRACKING

With the help of the LMS, the initialization problem of the BS-KF is drastically reduced since outliers can be filtered from the input vector before they are processed by the KF. To apply the LMS, K = 200 measurements are collected from each BS, of which only 11 values are randomly chosen for the LMS evaluation. Based on the given LMS estimate, the residuals for all measurements are computed and used to determine a weight w as follows

⎩⎨⎧ ≤

=otherwise,0

5.2|/][|if1][

skkw

r (17)

where k is the kth measurement in {1,…,K} and s is defined as (Rousseeuw and Leroy, 2003)

.}median{514826.1 rr o⎟⎟⎠

⎞⎜⎜⎝

⎛−

+=pK

s (18)

As mentioned before, application of the LMS delivers the most robust initialization and up to 50% of outliers can thus be handled. However, situations may arise where more outliers corrupt the measurements. This can lead to one of the following outcomes:

The LMS fails to estimate a linear fit and the program tries again after 100 new measurements.

The LMS succeeds achieving a linear fit, but this may be based on false measurements.

In the latter case, the KF must recognize the fault as soon as the proportion of correct measurements again exceeds 50% and initialize a track drop. This is the most critical case, since up to this point false data are forwarded to the position estimation algorithm.

c) CHI-SQUARED TEST FUNDAMENTALS A χ2 test is a well-known technique for detecting faulty measurements within a KF. The goal of this section is to describe the advantages of a χ2 test for tracking the output of a single BS over a χ2 test applied directly to the position solution. In the following, the necessary equations for performing a χ2 test are summarized, starting with the calculation of the information matrix of innovations Yk according to

(19) ( ,ˆ 1T1|

−−+= kkkkkk MSMRY )

with Rk as the covariance matrix of measurement errors and as the covariance matrix of the predicted state. The innovations νk themselves are given by the measurement residuals calculated by

1|ˆ −kkS

(20) ,ˆ 1| −−= kkkkk xMyν with as the predicted state according to the

model assumption. The χ2 value of a measurement set yk is given by

1|ˆ −kkx

kkkNνYνT2 1

=χ . (21)

Depending on the requirements of a specific application, it is possible to check either a single measurement with the χ2 test or even a complete data set at once. Sensor data are refused if the χ2 value exceeds a threshold level χthr

2, resulting in the test condition of . (22) 2

thr2 χχ >

According to (Grewal and Andrews, 2001) suitable threshold values should be determined by operational and not by theoretical values of χ2. For additional information concerning the χ2 distribution see (Papoulis, 1991).

d) CHI-SQUARED TEST FOR POSITION ESTIMATION MODE

As mentioned before, the χ2 test cannot distinguish between correct and faulty measurements before a correct state calculation is achieved. For a very limited number of faulty measurements a consistency check may work well. However, it is possible that the Kalman tracking filter is unable to converge to a suitable position solution even if sufficient BS measurements to calculate a precise position solution are left. Whereas for an EKF it is straightforward to screen each measurement separately by a χ2 test, the same procedure is computationally costly for a UKF.

e) CHI-SQUARED TEST FOR SINGLE BS TRACKING

Performing separate fault detections for each base station before a position solution is calculated has several big advantages over the previously discussed case. The resulting equations for the measurements and the state transition are linear. Furthermore, measurements of one BS do not influence another BS’s fault detection.

f) COMPLETE FAULT DETECTION The complete structure of the distributed fault detection for the LPM system is outlined in this subsection. A plot of the basic structure is shown in Fig. 3. Fault detection and robust filtering are achieved by an additional linear KF before the data transfer network sends the data to the MPU, which calculates a position solution for the MT relying on the raw measurements of each BS.

Figure 3 - Flow chart of the signals within the LPM processing chain. The proposed BS-KF will only forward pseudo-ranges which pass the χ2-test. The MPU provides the actual time tact as a counter variable. Robust filtering in the MPU can also be achieved by applying a χ2 test to all pseudo-ranges individually and directly at the EKF or UKF. A χ2 test is only available if the covariance matrix has reached a steady state. Until then the EKF has to rely purely on the unchecked pseudo-ranges and moreover has to be initialized with a good estimate that is adequately close to the true position. This is where the BS-KF has its greatest value: The pseudo-ranges are only forwarded to the initial position estimator and the EKF if they passed the χ2 test of the BS-KF. The additional χ2 test in the EKF is no longer necessary and can be skipped. Especially for a UKF-based state estimation this is of significant advantage as it reduces the calculation complexity. 5) MEASUREMENT RESULTS The BS-KF described in this contribution was tested in a real measurement scenario to lower the impact of

outliers in the position estimate. Outliers in position estimates may occur due to various reasons, such as:

Multi-path effects during the peak detection of the RT or MT.

Blockage of the line of sight (LOS) path and/or detection of the non-line-of-sight (NLOS) path.

Operation at the system’s limits, e.g., in vast measurement scenarios where the peaks of the RT and/or MT are close to the system noise floor.

a) TEST MEASUREMENTS

As a first step, the BS-KF was tested in a nearly perfect measurement environment. The basic measurement setup, which consisted of N = 12 BSs and an RT approximately in the middle of the BSs setup, is depicted in Fig. 4.

(a) Schematic of measurement setup

(b) Photography of measurement setup Figure 4 - Schematic and photograph of the measurement setup in front of the Abatec AG headquarter in Regau, Austria. The photo was taken at approximately position x = -5 m, y = -5 m looking in the direction of BS[9]. As shown in Fig. 5, the MT was mounted on an unmanned ground vehicle (UGV), which was performing nearly circular movements during measurement. Therefore, a CA movement model for the EKF was the proper choice as the undisturbed results show in Fig. 6.

Figure 5 - Photograph of the UGV equipped with a MT used to acquire the measurements. In Fig. 7, the output of the BS-KF of an arbitrarily chosen BS[n] during the first second is shown. Note that the pseudo-ranges display a high drift velocity (in this case about 1400 m/s) which obviously does not stem from the movement of the object but is caused by the different oscillator clocks of the RT and the MT. Due to this large increase of the pseudo-ranges over time, the master BS resets the start time of the MT, if the drift exceeds ±300 m. This reset leads to a jump in pseudo-ranges for which the algorithm has to compensate.

Figure 6 - Result of the EKF based on a CA movement model for the circular movement of the UGV. The outlier in the position estimate at x ≈ 3 m, y ≈ 0 m stems from the tuning phase of the EKF. During the first 0.35 s, the BS is in the initialization phase and the measurements are not forwarded to the MPU. After this phase, the KF is in track and the pseudo-ranges can be evaluated with the presented χ2 test. The output of this test is depicted in Fig. 8. Note that all χ2 values are far below the χ2

thr limit, which was set to 20.

Figure 7 - Plot of the measured pseudo-ranges (blue dots) of the first second for an arbitrarily chosen BS[n]. After 0.35 s the BS is initialized and the Kalman predictions become available (green circles).

Figure 8 - χ2 test results plotted over the entire measurement time. All values are below the threshold of 20, and therefore all measurements are accepted. In the next step, outliers were added to the measurements manually. Thus, (7) has to be reformulated to

},...1{},,...1{~]BS[][BS][BS MmNnew mnn ∈∈+= ρ

(22) where the bias eBS[m] is hereby derived from the maximum diameter of the given BS setup or ,

(23)

},...1{,||||max 21][BS][BS,

]BS[ 2121

Nmme mmmm

m ∈−= zz

where M is the number of simulated outliers at the BS with the index m. Theoretically, the number M of false measurements by the BSs can be up to N - r, where r is the dimension of the position estimation problem (e.g. r = 3 for 2D positioning) and the subsequent position estimator should nonetheless deliver a correct position estimate if all N - r false measurements can be excluded. Note that this would imply that the percentage of false data can be arbitrarily high as long as the number of BSs is

large enough. The critical drawback of this assumption is that neither the position of the (remaining) BSs nor the occurrence of the false data in time is taken into account. For a more detailed discussion on robustness in this context see (Rousseeuw and Leroy, 2003). In a first test, the pseudo-ranges of M = 5 randomly chosen BS were disturbed in each measurement cycle. As shown in Fig. 9 and 10, the BS-KF clearly now discards false measurements depending on their χ2 values.

Figure 9 - Plot of the corrupted pseudo-ranges (blue dots) and the estimates of the BS-KF (green circles) of the first second for an arbitrarily chosen BS. The measurements discarded by the χ2 test are marked with black diamonds.

Figure 10 - Plot of χ2 test results limited to a threshold of 20 of the first second for an arbitrarily chosen BS. After 0.35 s, the BS-KF starts processing and discards the corrupted data. In this test scenario, the χ2 values of the false measurement data are well separated from the true data. In Fig. 11 depicts a comparison between the position results of an unfiltered EKF and of an EKF with a prior BS-KF. While the EKF based on unfiltered data can only follow the movement of the UGV approximately, the EKF with the prior BS-KF delivers the same result as the original, undisturbed measurements.

(a) Position result in x direction

(b) Position result in y direction Figure 11 - Comparison of the position results of the EKF with (black solid line) and without (grey, dashed line) a prior BS-KF with outliers at five randomly chosen BS in each measurement cycle. The green dashed line marks the results without any corrupted data. Note that M = 5 marks the limit of for the manageable amount of outliers within one measurement set for the given measurement setup. In fact, any value of M > 5 would be equivalent to 50% or more outliers and would therefore signify a breakdown point > 50%. In (Rousseeuw and Leroy, 2003), a mathematically proof is provided that no estimator exists with a breakdown point > 50%, which also holds for the BS-KF.

b) FIELD MEASUREMENTS In this section, the performance of the BS-KF is demonstrated using measurements from an actual application of the LPM system. As mentioned in the introduction, a common application of the LPM system is to track soccer players on the field during their training. Here we present a recent example of such a player tracking at the training site of the Bayern Munich soccer club. As depicted in the photographs of Fig. 12, the training site is surrounded on three sides by a metal fence. If the LOS path to a specific BS is blocked (e.g. by the player’s head), the BS might still receive a signal reflected by the metal fence and deliver completely false data.

(a) Photograph of training area (left side)

(b) Photograph of training area (right side)

(c) Schematic of measurement setup Figure 12 - Photographs and schematic from the measurement setup at the Bayern Munich soccer club training site in Munich, Germany. The photographs were taken around the position of BS[1] facing BS[4] and BS[7] respectively. To analyse the system performance, a walk along the edges of the soccer field was recorded and evaluated with an EKF using an RW model for the movement. In Fig. 13, the position estimation results of the EKF are depicted with (EKF1) and without (EKF2) a prior BS-KF. For better comparability, neither EKF uses an additional χ2 test, which would improve the results of EKF2. Note that EKF1 has almost no tuning phase since at initialization time all estimation parameters are known including the drift velocity of the measurement offset.

The great advantage of applying the BS-KF becomes apparent for values of t between 70 and 90 seconds (approximately). While EKF2 accepts false data within the measurements and therefore delivers false position estimates, the BS-KF protects EKF1 from false data. Unfortunately, only 3 BSs are then left, which keeps EKF1 from updating its state estimate and so EKF1 can only outputs its predicted values. The RW model causes the EKF1 to predict that the object remains static, which can be seen in the steps of the position estimate in y-direction. Nevertheless, this prediction is far more appropriate than the estimate of EKF2.

(a) Position result in x direction

(b) Position result in y direction

(c) Position result plotted x against y Figure 13 - Comparison of the position results of the EKF with (black solid line) and without (grey, dashed line) a prior BS-KF in x (a), and y (b) direction plotted against time and x plotted against y (c)

respectively. In the latter case, the labels of the BSs and RT are omitted. 6) CONCLUSION In this contribution a new IE for the LPM system is presented, which monitors the measured data for coarse, false data. The applicability of the presented distributed fault detection strategies is demonstrated using real measurements. It was shown that the BS-KF can successfully discard false pseudo-ranges, which confirms that the complete system is robust against outliers. Furthermore, the BS-KF performs in real-time and does not create any drawback if the measurements include no false data. 7) ACKNOWLEDGMENTS Part of this work was funded by the COMET K2 Center “Austrian center of competence in mechatronics” (ACCM). The COMET Program is funded by the Austrian Federal government, the Federal State Upper Austria and the Scientific Partners of ACCM. The authors would like to thank Andreas Resch and Stephan Reinsch from Abatec Electronic AG for their continuous support. 8) REFERENCES Anderson, B. D. O. and Moore, J. B. (1979). Optimal Filtering, Englewood Cliffs, NJ: Prentice-Hall. Blackman, S. and Popoli, R. (1999). Design and Analysis of Modern Tracking Systems. Norwood: Artech House. Brookner, E. (1998). Tracking and Kalman Filtering Made Easy, New York: John Wiley & Sons. Grewal, M. S. and Andrews, A. P. (2001). Kalman Filtering: Theory and Practice using Matlab. 2nd ed. New York, NY: John Wiley & Sons. Li, X. R. and Jilkov, V. P. (2003). Survey of Maneuvering Target Tracking Part I: Dynamic Models. In IEEE Transactions on Aerospace and Electronic Systems, vol. 39, no. 4, pp. 1333-1364. Mitchell, H. B. (2007). Multi-Sensor Data Fusion. Berlin: Springer. Papoulis, A. (1991). Probability, Random Variables, and Stochastic Processes. 3rd ed. New York: NY: McGraw-Hill.

Pfeil, R., Schuster, S., Scherz, P., Stelzer, A. and Stelzhammer, G. (2009). A Robust Position Estimation Algorithm for a Local Positioning Measurement System. In Proceedings of the International Microwave Workshop Series on Wireless Sensing, Local Positioning, and RFID (IMWS), Cavtat, Croatia, Sep. 24-25. Pourvoyeur, K., Stelzer, A. and Gassenbauer, G. (2006). Position Estimation Techniques for the Local Position Measurement System LPM. In Asian-Pacific Microwave Conference (APMC), Yokohama, Japan, Dec. 12-15. Pourvoyeur, K. (2007). Position Estimation, Tracking, and Sensor Fusion of the Local Position Measurement System LPM, PhD-Thesis (204 pages), Johannes Kepler University Linz. Pourvoyeur, K., Stelzer, A., and Stelzhammer, G., (2008). Error Estimation for Reliable Fault Detection of a TDOA Local Positioning System. In Proc. Tyrrhenian International Workshop on Digital Communications / Enhanced Surveillance of Aircraft

and Vehicles (TIWDC/ESAV) on CD-ROM (6 pages), Sept. 3–5, Capri, Italy. Rousseeuw, P. J. and Leroy, M. (2003). Robust Regression and Outlier Detection. Hoboken: John Wiley & Sons. Simon, D. (2006). Optimal State Estimation, New York, NY: John Wiley & Sons. Skolnik, M. I. (2001). Introduction to Radar Systems. 3rd ed. Boston, MA: McGraw-Hill. Stelzer, A., Pourvoyeur, K. and Fischer, A. (2004). Concept and Application of LPM – A novel 3-D local position measurement system. In IEEE Transactions on MTT-S, vol. 52 no. 12, Dec 2004. Stove, A. G. (1992). Linear FMCW Radar Techniques. In IEE Proc. F, Radar Signal Processing, vol. 139(5), pp. 343-350.